High eccentricity ``spike'' in the multiple planet population

 I seek collaborators to put the following discovery into my paper in preparation on features in the number distribution and eccentricity of the exoplanet population. This is a newer discovery of a ``spike'' in the eccentricity distribution by period of the planets in multiple planet systems. I add this to my finding of the double peak around a gap and my finding of the nature of there being a correlation between eccentricity and iron abundance of the star that changes with period. This includes how there is a region in between 500 and 600 days where the eccentricity of orbits of stars more iron-poor than the sun ``spikes'' in eccentricity.

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Introduction

How could there be a ``spike’’ of five high eccentricities clumped together in period out of one population of planets hosted by sunlike stars that has 95 planets selected to be sunlike in temperature, surface gravity, and absence of a stellar companion? Specifically, when the eccentricities versus period of the population of just planets that are in multiple planets systems, five eccentricities are clumped together in period that collectively have higher eccentricities than anywhere elsewhere in the logarithmic period space of exoplanets. These eccentricities especially stand out above the rest when looking at the 20 planets with periods from 10 to 100 days by being much higher than any eccentricities of the other 15 planets. Though among these 95 planets there is a rise in the spread of eccentricities with increasing period, four of these five eccentricities are still higher than any of the other 90 eccentricities.

Fig. 1. The eccentricities versus periods of all stars. The eccentricities of orbits of stars more iron abundant than the sun (``iron-rich,'' red open circles) are higher at most all periods than the eccentricities of orbits of stars less iron abundant than the sun (` `iron-poor,'' blue filled circles).

Fig. 2. Eccentricities of the orbits of planets around only stars that are ``sunlike'' in temperature and surface gravity, and in being single stars. Symbols same as Fig. 1. The lower eccentricity at most periods of orbits of stars more iron rich than the sun can be seen, as well as the peaking of the eccentricity of orbits of stars that are poor in iron relative to the sun at periods above 500 days.

 

 Eccentricity as a function of period for selected populations

All and Sunlike:

The eccentricity as a function of period, with whether iron-abundance is poorer or richer than solar indicated, are shown in Fig. 1 for the full and some selections of the 429 orbits of planets found by radial velocity (RV) found with periods of up to 5000 days, followed by Fig. 2 which shows the selection of 243 orbits chosen for stars that are more ``sunlike’’. The ``sunlike’’ sample was selected by taking the 243 planets of the 429 available objects found by RV, where ``objects’’ refers to the set of parameters describing a planet, its star, and their orbit. Stars with different parameters might not have the same features or have them at the same period, so only stars similar to the sun are compared here. Since planet searches have emphasized sunlike stars, this group of stars has the highest number of objects with similar stars available for study. Sunlike objects are those which have stars that have no stellar companion with effective temperatures (or Teff) of 4500 to 6500 K to be close to the Teff of the sun of 5772 K, and with surface gravity not too much less than the sun's value given in logarithmic terms of 4.4. We do not at this time remove stars with very different masses than the sun because not too many remain in this sample, but it may later be important since the small data on lower mass stars could indicate that the peaks and gap feature may occur at shorter periods. Different markers are used to separate orbits by whether the star is poorer or richer in iron abundance than the sun, [Fe/H] <= 0 or [Fe/H] > 0 respectively, shown by the blue filled or red unfilled circles respectively. Table 1 gives the counts in each cut with each cut divided into how many iron-poor and iron-rich objects there are.

Features stand out:

Several features stand out, starting with a broad increase in eccentricity with period at the shorter periods that results from the shortest period orbits having their eccentricities reduced (commonly said to be ``circularized'') due to tidal interaction with the star.  This affects all populations of orbits. When those orbits of stars that have less or more iron than the sun are separated, which are referred to as ``iron-poor'' and ``iron-rich'' objects, it can be seen that the eccentricities of the iron-rich objects rise more rapidly from zero than do the eccentricities of the iron-poor objects, leading to the ``eccentricity-metallicity’’ correlation found between the eccentricity of moderately short period planet orbits and iron abundances found by Taylor (2012, 2013b) and Dawson \& Murray-Clay (2013). This correlation is strongest at periods of roughly 100 days (the ``valley’’ region) but may persist more weakly at periods up to 500 days. The eccentricities of the iron-rich objects have a broader peak, while the eccentricities of the iron-poor objects come more sharply to a peak, and then decline more. The result of the peaking of the eccentricities of iron-poor objects is that the correlation between eccentricity and iron abundance goes away for a middle range of periods from 500 days (shown in Taylor 2013b) upward into the periods where we are showing there is a gap in the number distribution of iron-rich objects. We are preparing work that shows that beyond that gap, the correlation likely returns.

In further work, it will be shown that the correlation of eccentricity is not simply bimodal with iron abundance but the eccentricity changes gradually with iron abundance, that is that the eccentricities of objects slightly above solar have, in periods where the correlation exists, a higher means than iron-poor objects but lower means than for objects with the highest iron abundances.

Fig. 3. Eccentricity versus period of planet orbits of sunlike stars in single-planets, with marker symbols showing whether the iron abundance is above or below solar ([Fe/H] of 0) as in Fig. 1.

Fig. 4. The ''spike'' in the eccentricity of planet orbits of sunlike stars that are in multiple planets systems can be seen clearly here all being between periods of 44 and 75 days. Symbols as in Fig 1. Four of these five are higher than the eccentricity of any other eccentricity. The eccentricity clearly slowly rises with period for both iron-poor and iron-rich stars, though few iron-poor stars are found in multiple systems at longer periods.

Different patterns in eccentricity by period of orbits

 in single-planet versus Multi-planet systems:

In the next two figures the eccentricity versus period is shown for two populations formed by further dividing the sunlike sample of 243 objects. Fig. 3 shows the eccentricities for the 148 ``single-planet’’ objects for which the planet is the only planet found orbiting its host star, and Fig. 4 shows eccentricities for the 95 ``multi-planet’’ objects for which one or more additional planets has been found.

The single-planet sample has a distribution that retains more of the description of the full sample, as this sample has higher mean eccentricities at most periods. This is expected given that the orbits in multi-planet systems are constrained from being too eccentric.

The eccentricities of the multi-planet sample do not peak but continue to rise with period, though the number counts of iron-poor multi-planet objects drops off such that there are far fewer iron-poor multi-planet objects than iron-rich multi-planet objects at periods longer than 1000 days.For multiplanet objects in just sunlike systems there is 1 iron poor vs 23 iron-rich objects at periods longer than 1000 days.

Single-planet and Multi-planet:

We take the 243 sunlike objects separately show the eccentricity versus period distributions for the 148 orbits of ``single-planet’’ objects, or of planets that are the only planet found, and of 95 ``mutiple-planet’’, or of planets for which at least one more planetary companion has been found. The breakdown of counts into iron-poor and rich objects are given in Table 1.

The two distributions look quite different, with the eccentricities of the multiple planets lower in general. This is as expected that a planet is more likely to have a more orderly orbit if there is another planet in the system. The full description of how different these two populations are will be given in upcoming work, while the focus here is on the spike in eccentricity in the multiple planet population. Some qualitative differences besides the spike jump out, including how in the multiple-planet population, the number of iron-poor objects drops off at longer periods. This drop off contributes to the ratio of iron-poor to rich objects being higher for the single planets (40:108 or 0.37) than for multiple planets (23:72 or 0.32).

Whether other than the spike the shape of the eccentricity distributions in the multiple planet population bear a lower eccentricity resemblance to the single planet distribution is a subject of current work. The highest eccentricity point of the iron-poor multiple planet population, HD_192310_c, is at 0.32 much higher than the 2^nd highest eccentricity of 0.21 (the 24.451 day HD_7924_d). Its 525.8 ± 9.2 day period fits within the 500 to 600 day period range of the spike in eccentricity of the general population that is the subject of Taylor (2014). While it may be difficult to attach too much significance to one point, it is notable for being such an outlier.

The region of the spike in the eccentricity versus period distribution of the multiple planet population corresponds to a region without similarly high eccentricity objects in the same period range of the single planet population, though if objects that include stars with stellar companions are not cut, there is one such object in this period range of the iron-rich population of the full (not sunlike) selection.

This does raise the possibility that perhaps the spike is simply ``cut out’’ of the single plus multiple planet population by a greater likelihood of finding a planetary companion for planets within this region. This could be true of either physical causes or observational effects. It seems unlikely, though, that the shorter period edge could be a result of not finding companions to shorter period planets. It is worth further work studying this possible effect.

Table 1. Counts of numbers of objects in figures in each cut, with counts of objects divided by whether the abundance of iron of the star is poorer or richer than the sun.

Fig Number	Selection	Total objects	Iron-poor objects	Iron-rich objects
1	All RV objects	429	149	280
2	``Sunlike’’ objects in temperature and surface gravity	243	63	180
3	Single-planet sunlike objects	148	40	108
4	Multiple-planet sunlike objects	95	23	72

Spike Description:

The five planets comprising this spike have orbits with periods from 44 .2 to 75.3 days. Planets orbits tend to be spaced at increasing distances such that it is best to look at planet orbits in ``log space’’ where it is common to give the logarithm to the base 10. When looked at in log period space, this is the very small range of 0.23, going from 1.65 to 1.88 in logarithmic period. This 0.23 is very small given that the range of RV planets that are comparable can be said to have a length of 2.58 in ``log period space’’, going from below 10 days to 5000 days, which is going from log of 1 to log of 3.70. How could planets that are in multiple planet systems have orbits with the highest eccentricities be confined to such a small range?

We must evaluate not only whether this spike could be random or observational, but also whether it could be a result of making the selection of the parameters, especially on making the selection on multiple planet systems. We address how could be possible that planets selecting a part of a larger population, in this case choosing those planets that are in multiple planet versus single planet systems, could have led to a small range in period of planets being preferentially put into this multiple-planet population while planets just outside this range might be preferentially chosen into the single-planet population. It could be the observational effect that planets in slightly longer periods would actually still be in multiple systems but at longer periods that are still too long for them to have been found. A similar explanation would be that there simply are not further planets at the longer periods, but this physical explanation would be of interest as it would be relevant to the existence of peaks in the planet population counted by period.

The presence of high eccentricity orbits in the corresponding period range shortwards of the spike in the single planet system population argues against explanation of tidal dissipation in the star creating the short-period edge of the spike. It also argues against  not having found companions as an observational effect since these companion would not be expected to have longer periods, unless there is a physical reason for the companions not to have been found. It is possible that the longer period companions would have periods within the gap that has been found in the iron-rich population, but that these companion planets are simply ``not present’’ due to the gap.`

Likelihood Section:

Calculations show a low likelihood that high eccentricity orbits would be so close in period

The first questions to ask when seeing an apparent feature must be to determine whether the feature is a real physical feature, starting with asking if the feature might just be a random fluctuation in a small numbers of data points.

The chance that the highest planets in eccentricity would be confined such a small range depends on whether the likelihood is evaluated as being the chance of five high values occurring in the somewhat local range where all five are the highest values, or if the chance of the four highest values occurring over the entire range, but the two give similar results of under one percent and a few thousandths respectively.

The five orbits can be considered to by five pairs of log period and eccentricity values. These are listed below, with the (not log) periods in days preceding each pair for easy reference:

Table 2: Period 

(in days and in base 10 logarithm of the period in days) 

and eccentricity of the five high eccentricity objects.

PER	log period	ECC
day
44.24	1.65	0.47
51.64	1.71	0.63
55.01	1.74	0.68
58.11	1.76	0.53
75.29	1.88	0.73

The calculated likelihood of a certain number of periods occurring within a larger range depends on the length of the range we consider these values might have occurred in. Below we will calculate the likelihood for the five values to be higher in a large part of the full range in period, and then will calculate the likelihood that four values have higher eccentricity than any other object at any period. We choose to be conservative by considering that higher periods are only likely to be larger for longer period orbits, due to the general pattern for eccentricities to get larger with increasing period as a general pattern up to periods of several hundred days. We consider that the eccentricities of shorter period orbits may be lessened by the tidal interaction with the star that tends to circularize the shortest period orbits, so to be conservative, we look for the possibility that these eccentricities randomly occur at some period from shortest period of the spike to one of two longer periods discussed below. It should be noted that in the population of planets without planetary companions, there are high-eccentricity planets by periods of 20 days, so the period ranges given below could have been taken to be longer, even further reducing the likelihoods given below. Looking at the values of eccentricity versus period for single planets shows higher eccentricities for shorter periods in that population, however, leading the values calculated here to give a higher likelihood of this spike resulting from chance, but we choose to err on the conservative side.  In calculating the probability that these periods will occur within the spike range, we take the shortest period of the high eccentricity points, 44.2 days or 1.65 in log-period space, as the shortest period of the range of the periods just as likely to have high eccentricity orbits. The next period at which a higher eccentricity than the lowest of the five occurs is at 567.9 days, or 2.75 in log period space, so these five values could have occurred anywhere along a log period range of 1.11, but they all occurred within 0.23. So the likelihood that five points that could have occurred in 1.11 but occurred in 0.23 can be calculated by finding how often five values appear in the fraction 0.23/1.11=0.21 of points randomly generated from 0 to 1.

Performing random selections of one million sets of five periods from 0 to one shows that only 0.7% of random selections of five values will be within a range of 0.21 of each other. We calculate this by taking the difference between the highest and lowest of the five selected values to allow for the possibility of the five points grouping anywhere within this range. We repeat this procedure to consider how likely is it that the highest four points are within the short range that we find them where we could find them in anywhere up to the full 5000 days for which RV periods are available, or 1.99 in log period from the log values of the period range of 51.6 days to 5000 days. Since the lowest of the five eccentricity values also corresponds to the shortest period, there are now four points from periods 51.6 to 75.3 days, which is 1.71 to 1.88 in log period, spanning a range of 0.16 in log period. This is 0.082 of the 1.99. The chance of randomly having four points within 0.082 randomly generated from points from 0 to 1 is 0.21%. We conclude that this feature is unlikely due to random clustering of the periods at the better than 1% level.

Abundance

The five high eccentricity orbits are characterized by much higher iron abundance in the stars than in the other 15 of the 20 orbits in the full population in the similar period range from 10 to 100 days, reflecting the strong correlation between iron-abundance and eccentricity found in this range. Four of the five have iron abundances higher than all 15 of the stars with low eccentricity planets in this range, and the lowest iron abundance of the five is still higher than more than half of the other 15.

Discussion

Simulations of the likelihood that the periods of the four or five highest eccentricity objects show that the spike is unlikely to be completely random. It is also unlikely to be completely observational effect, but the presence of high eccentricities in a small range of period could be influenced by the interaction between the physical distribution and how additional planets are found.

It is essential to consider whether this spike is merely ``shaved’’ out of the full distribution. The short and long period edges are considered separately. The shorter period edge of the spike could be created by shorter-period orbits having their eccentricities reduced by tidal dissipation. For the longer period edge, there is the possibility that there are higher eccentricity longer-period orbits of planets in systems with more than one planet for which the additional planets have not yet been found so these eccentricity values are still showing in the single-planet plot instead. While both of these possibilities should be researched further, comparison with the single-planet distribution gives some evidence that these are not the explanation for the appearance of a spike. This evidence includes how orbital circularization falls off more quickly allowing higher eccentricities in the single-planet population with periods shorter than 44 days. The single-planet distribution has a rise in eccentricities for periods much shorter than 44 days, and it is more of a gradual rise. It must be noted that the region of the spike shows a possible hole in this region in period space of the high-eccentricity envelope of the single-planet population.

At slightly longer periods, there is a paucity of high-eccentricity systems for both single and multiple planet populations from the range of the spike to over 100 day periods, so there are not enough values there to move over to the multi-planet population. (It is worth noting this paucity that is longwards of the spike, but low statistics makes it uncertain that this paucity is an actual gap feature.)

Does having so many different sources invalidate these results?

It is important to address that the catalog of planet orbits is collected from many different surveys which can have very different efficiencies and standards, which lead many to distrust looking in such a collected dataset for patterns. It is hard to believe that the appearance of these features could be from differences between different observers, especially given how different populations show clearly different patterns that it is improbable that observers could be selecting for. Any observational effects should similarly affect measurements of planets hosted by sunlike or not sunlike stars, and single or multiple planets or stars. For some features to appear so strongly should give confidence that the quality of RV exoplanet data is consistently very high.

Further observations to lead to further work:

It appears that, other than the spike, the shape of eccentricity distribution by period of iron-poor and iron-moderately-high multiple planets appears to be a ``pushed down’’ version of the single planets distribution. Future work must address whether both the iron poor objects in the full population and the iron poor objects in the multiple planet population have a similar spreading of the spike with increasing iron abundance in the range  0 < [Fe/H] < 0.1 of the in eccentricity that occurs in the 500-600 day range.

Conclusion:

The finding of a spike in eccentricity in the population of planets with planetary companions again shows that pattern formation and evolution leads to more uniform distributions than expected. The presence of distinct features in the eccentricity and number distribution by period shows that the evolution of planets which could include activity such as planet scattering after formation that could smooth out these patterns, is likely not to overly disturb patterns that occur in system after systems.

These results suggest that the pattern of planet formation is like more predictable and less random from one planet system to the next. The presence of these features prompts the suggestion that observers of protoplanetary disks (PPDs) look for whether the rings and gaps in PPDs tend to have repeating patterns from disk to disk, or if the features now being found in PPDs tend to be found at random periods. The preservation of features in the number and eccentricity distribution presents the opportunity to learn about planet formation through studying features found in the parameters of mature planet systems.